A319254 -id:A319254 - cp's OEIS frontend

A000669 Number of series-reduced planted trees with n leaves. Also the number of essentially series series-parallel networks with n edges; also the number of essentially parallel series-parallel networks with n edges.

1, 1, 2, 5, 12, 33, 90, 261, 766, 2312, 7068, 21965, 68954, 218751, 699534, 2253676, 7305788, 23816743, 78023602, 256738751, 848152864, 2811996972, 9353366564, 31204088381, 104384620070, 350064856815, 1176693361956, 3963752002320

Offset: 1

N. J. A. Sloane and John Riordan

Also the number of unlabeled connected cographs on n nodes. - N. J. A. Sloane and Eric W. Weisstein, Oct 21 2003
A cograph is a simple graph which contains no path of length 3 as an induced subgraph. - Michael Somos, Apr 19 2014
Also called "hierarchies" by Genitrini (2016). - N. J. A. Sloane, Mar 24 2017

			G.f. = x + x^2 + 2*x^3 + 5*x^4 + 12*x^5 + 33*x^6 + 90*x^7 + 261*x^8 + ...
a(4)=5 with the following series-reduced planted trees: (oooo), (oo(oo)), (o(ooo)), (o(o(oo))), ((oo)(oo)). - _Michael Somos_, Jul 25 2003

N. L. Biggs et al., Graph Theory 1736-1936, Oxford, 1976, p. 43.
A. Brandstaedt, V. B. Le and J. P. Spinrad, Graph Classes: A Survey, SIAM Publications, 1999. (For definition of cograph)
A. Cayley, Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 3, p. 246.
D. E. Knuth, The Art of Computer Programming, 3rd ed. 1997, Vol. 1, p. 589, Answers to Exercises Section 2.3.4.4 5.
L. F. Meyers, Corrections and additions to Tree Representations in Linguistics. Report 3, 1966, p. 138. Project on Linguistic Analysis, Ohio State University Research Foundation, Columbus, Ohio.
L. F. Meyers and W. S.-Y. Wang, Tree Representations in Linguistics. Report 3, 1963, pp. 107-108. Project on Linguistic Analysis, Ohio State University Research Foundation, Columbus, Ohio.
J. Riordan and C. E. Shannon, The number of two-terminal series-parallel networks, J. Math. Phys., 21 (1942), 83-93 (the numbers called a_n in this paper). Reprinted in Claude Elwood Shannon: Collected Papers, edited by N. J. A. Sloane and A. D. Wyner, IEEE Press, NY, 1993, pp. 560-570.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

N. J. A. Sloane, First 1001 terms of A000669
Mohamed Barakat, Reimer Behrends, Christopher Jefferson, Lukas Kühne, and Martin Leuner, On the generation of rank 3 simple matroids with an application to Terao's freeness conjecture, arXiv:1907.01073 [math.CO], 2019.
Peter J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
Peter J. Cameron, Some treelike objects, Quart. J. Math. Oxford, 38 (1987), 155-183. MR0891613 (89a:05009). See pp. 155, 162, 165, 172. - _N. J. A. Sloane_, Apr 18 2014
Peter J. Cameron, Some sequences of integers, Discrete Math., 75 (1989), 89-102; also in "Graph Theory and Combinatorics 1988", ed. B. Bollobas, Annals of Discrete Math., 43 (1989), 89-102.
Maria Chudnovsky, Jan Goedgebeur, Oliver Schaudt, and Mingxian Zhong, Obstructions for three-coloring graphs without induced paths on six vertices, arXiv preprint arXiv:1504.06979 [math.CO], 2015-2018.
Steven R. Finch, Series-parallel networks
Steven R. Finch, Series-parallel networks, July 7, 2003. [Cached copy, with permission of the author]
Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018.
Philippe Flajolet, A Problem in Statistical Classification Theory
P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 72
Daniel L. Geisler, Combinatorics of Iterated Functions
A. Genitrini, Full asymptotic expansion for Polya structures, arXiv:1605.00837 [math.CO], May 03 2016, p. 9.
O. Golinelli, Asymptotic behavior of two-terminal series-parallel networks, arXiv:cond-mat/9707023 [cond-mat.stat-mech], 1997.
JiSun Huh and Seonjeong Park, Toric varieties of Schröder type, arXiv:2204.00214 [math.AG], 2022. (Table 1)
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 44 [broken link].
V. P. Johnson, Enumeration Results on Leaf Labeled Trees, Ph. D. Dissertation, Univ. Southern Calif., 2012. - From _N. J. A. Sloane_, Dec 22 2012
P. A. MacMahon, Yoke-trains and multipartite compositions in connexion with the analytical forms called "trees", Proc. London Math. Soc. 22 (1891), 330-346; reprinted in Coll. Papers I, pp. 600-616. Page 333 gives A000084 = 2*A000669.
Arnau Mir, Francesc Rossello, and Lucia Rotger, Sound Colless-like balance indices for multifurcating trees, arXiv:1805.01329 [q-bio.PE], 2018.
V. Modrak and D. Marton, Development of Metrics and a Complexity Scale for the Topology of Assembly Supply Chains, Entropy 2013, 15, 4285-4299.
J. W. Moon, Some enumerative results on series-parallel networks, Annals Discrete Math., 33 (1987), 199-226.
Vlady Ravelomanana and Loys Thimonier, Asymptotic enumeration of cographs, Electronic Notes in Discrete Mathematics, Volume 7, April 2001, pp. 58-61, Theorem 4.
J. Riordan, The blossoming of Schröder's fourth problem, Acta Math., 137 (1976), 1-16. (page 6)
J. Riordan, Letter to N. J. A. Sloane, Sep. 1970
J. Riordan, Letter to N. J. A. Sloane, Nov 10 1970
Audace Amen Vioutou Dossou-Olory, and Stephan Wagner, Inducibility of Topological Trees, arXiv:1802.06696 [math.CO], 2018.
Audace Amen Vioutou Dossou-Olory, The topological trees with extremal Matula numbers, arXiv:1806.03995 [math.CO], 2018.
Wei Wang and Ximei Huang, Almost all cographs have a cospectral mate, arXiv:2507.16730 [math.CO], 2025. See pp. 6, 8.
Eric Weisstein's World of Mathematics, Series-Parallel Network
Index entries for sequences related to rooted trees
Index entries for sequences mentioned in Moon (1987)
Index entries for sequences related to trees

Equals (1/2)*A000084 for n >= 2.
Cf. A000055, A000311, A001678, A007827, A363064.
Cf. A000311, labeled hierarchies on n points.
Column 1 of A319254.
Main diagonal of A292085.
Row sums of A292086.

Method 1: a := [1,1]; for n from 3 to 30 do L := series( mul( (1-x^k)^(-a[k]),k=1..n-1)/(1-x^n)^b, x,n+1); t1 := coeff(L,x,n); R := series( 1+2*add(a[k]*x^k,k=1..n-1)+2*b*x^n, x, n+1); t2 := coeff(R,x,n); t3 := solve(t1-t2,b); a := [op(a),t3]; od: A000669 := n-> a[n];
Method 2, more efficient: with(numtheory): M := 1001; a := array(0..M); p := array(0..M); a[1] := 1; a[2] := 1; a[3] := 2; p[1] := 1; p[2] := 3; p[3] := 7;
Method 2, cont.: for m from 4 to M do t1 := divisors(m); t3 := 0; for d in t1 minus {m} do t3 := t3+d*a[d]; od: t4 := p[m-1]+2*add(p[k]*a[m-k],k=1..m-2)+t3; a[m] := t4/m; p[m] := t3+t4; od: # A000669 := n-> a[n]; A058757 := n->p[n];
# Method 3:
b:= proc(n, i) option remember; `if`(n=0, 1,
      `if`(i<1, 0, add(binomial(a(i)+j-1, j)*
       b(n-i*j, i-1), j=0..n/i)))
    end:
a:= n-> `if`(n<2, n, b(n, n-1)):
seq(a(n), n=1..40);  # Alois P. Heinz, Jan 28 2016

b[n_, i_] := b[n, i] = If[n==0, 1, If[i<1, 0, Sum[Binomial[a[i]+j-1, j]* b[n-i*j, i-1], {j, 0, n/i}]]];
a[n_] := If[n<2, n, b[n, n-1]];
Array[a, 40] (* Jean-François Alcover, Jan 08 2021, after Alois P. Heinz *)

{a(n) = my(A, X); if( n<2, n>0, X = x + x * O(x^n); A = 1 / (1 - X); for(k=2, n, A /= (1 - X^k)^polcoeff(A, k)); polcoeff(A, n)/2)}; /* Michael Somos, Jul 25 2003 */

from collections import Counter
def A000669_list(n):
    list = [1] + [0] * (n - 1)
    for i in range(1, n):
        for p in Partitions(i + 1, min_length=2):
            m = Counter(p)
            list[i] += prod(binomial(list[s - 1] + m[s] - 1, m[s]) for s in m)
    return list
print(A000669_list(20)) # M. Eren Kesim, Jun 21 2021

Product_{k>0} 1/(1-x^k)^a_k = 1+x+2*Sum_{k>1} a_k*x^k.
a(n) ~ c * d^n / n^(3/2), where d = 3.560839309538943329526129172709667..., c = 0.20638144460078903185013578707202765... [Ravelomanana and Thimonier, 2001]. - Vaclav Kotesovec, Aug 25 2014
Consider a nontrivial partition p of n. For each size s of a part occurring in p, compute binomial(a(s)+m-1, m) where m is the multiplicity of s. Take the product of this expression over all s. Take the sum of this new expression over all p to obtain a(n). - Thomas Anton, Nov 22 2018

Sequence crossreference fixed by Sean A. Irvine, Sep 15 2009

A050381 Number of series-reduced planted trees with n leaves of 2 colors.

Original entry on oeis.org

2, 3, 10, 40, 170, 785, 3770, 18805, 96180, 502381, 2667034, 14351775, 78096654, 429025553, 2376075922, 13252492311, 74372374366, 419651663108, 2379399524742, 13549601275893, 77460249369658, 444389519874841

Offset: 1

Christian G. Bower, Nov 15 1999

nonn

Consider the free algebraic system with two commutative associative operators (x+y) and (x*y) and two generators A,B. The number of elements with n occurrences of the generators is 2*a(n) if n>1, and the number of generators if n=1. - Michael Somos, Aug 07 2017
From Gus Wiseman, Feb 07 2020: (Start)
Also the number of semi-lone-child-avoiding rooted trees with n leaves. Semi-lone-child-avoiding means there are no vertices with exactly one child unless that child is an endpoint/leaf. For example, the a(1) = 2 through a(3) = 10 trees are:
  o    (oo)      (ooo)
  (o)  (o(o))    (o(oo))
       ((o)(o))  (oo(o))
                 ((o)(oo))
                 (o(o)(o))
                 (o(o(o)))
                 ((o)(o)(o))
                 ((o)(o(o)))
                 (o((o)(o)))
                 ((o)((o)(o)))
(End)

			For n=2, the 2*a(2) = 6 elements are: A+A, A+B, B+B, A*A, A*B, B*B. - _Michael Somos_, Aug 07 2017

Andrew Howroyd, Table of n, a(n) for n = 1..500
David Callan, A sign-reversing involution to count labeled lone-child-avoiding trees, arXiv:1406.7784 [math.CO], (30-June-2014).
F. Chapoton, F. Hivert, J.-C. Novelli, A set-operad of formal fractions and dendriform-like sub-operads, arXiv preprint arXiv:1307.0092 [math.CO], 2013.
V. P. Johnson, Enumeration Results on Leaf Labeled Trees, Ph. D. Dissertation, Univ. Southern Calif., 2012. - From _N. J. A. Sloane_, Dec 22 2012
N. J. A. Sloane, Transforms
Gus Wiseman, Sequences counting series-reduced and lone-child-avoiding trees by number of vertices.
Index entries for sequences related to rooted trees

Column 2 of A319254.
Cf. A029856, A031148.
Lone-child-avoiding rooted trees with n leaves are A000669.
Lone-child-avoiding rooted trees with n vertices are A001678.
The locally disjoint case is A331874.
Semi-lone-child-avoiding rooted trees with n vertices are A331934.
Matula-Goebel numbers of these trees are A331935.
Cf. A000081, A005804, A141268, A196545, A291636, A316697, A330465, A331872, A331933, A331964.

terms = 22;
B[x_] = x O[x]^(terms+1);
A[x_] = 1/(1 - x + B[x])^2;
Do[A[x_] = A[x]/(1 - x^k + B[x])^Coefficient[A[x], x, k] + O[x]^(terms+1) // Normal, {k, 2, terms+1}];
Join[{2}, Drop[CoefficientList[A[x], x]/2, 2]] (* Jean-François Alcover, Aug 17 2018, after Michael Somos *)
slaurte[n_]:=If[n==1,{o,{o}},Join@@Table[Union[Sort/@Tuples[slaurte/@ptn]],{ptn,Rest[IntegerPartitions[n]]}]];
Table[Length[slaurte[n]],{n,10}] (* Gus Wiseman, Feb 07 2020 *)

{a(n) = my(A, B); if( n<2, 2*(n>0), B = x * O(x^n); A = 1 / (1 - x + B)^2; for(k=2, n, A /= (1 - x^k + B)^polcoeff(A, k)); polcoeff(A, n)/2)}; /* Michael Somos, Aug 07 2017 */

Doubles (index 2+) under EULER transform.
Product_{k>=1} (1-x^k)^-a(k) = 1 + a(1)*x + Sum_{k>=2} 2*a(k)*x^k. - Michael Somos, Aug 07 2017
a(n) ~ c * d^n / n^(3/2), where d = 6.158893517087396289837838459951206775682824030495453326610366016992093939... and c = 0.1914250508201011360729769525164141605187995730026600722369002... - Vaclav Kotesovec, Aug 17 2018

A319539 Array read by antidiagonals: T(n,k) is the number of binary rooted trees with n leaves of k colors and all non-leaf nodes having out-degree 2.

Original entry on oeis.org

1, 2, 1, 3, 3, 1, 4, 6, 6, 2, 5, 10, 18, 18, 3, 6, 15, 40, 75, 54, 6, 7, 21, 75, 215, 333, 183, 11, 8, 28, 126, 495, 1260, 1620, 636, 23, 9, 36, 196, 987, 3600, 8010, 8208, 2316, 46, 10, 45, 288, 1778, 8568, 28275, 53240, 43188, 8610, 98, 11, 55, 405, 2970, 17934, 80136, 232500, 366680, 232947, 32763, 207

Offset: 1

Andrew Howroyd, Sep 22 2018

Not all k colors need to be used. The total number of nodes will be 2n-1.

See table 2.1 in the Johnson reference.

			Array begins:
===========================================================
n\k|  1    2      3       4        5        6         7
---+-------------------------------------------------------
1  |  1    2      3       4        5        6         7 ...
2  |  1    3      6      10       15       21        28 ...
3  |  1    6     18      40       75      126       196 ...
4  |  2   18     75     215      495      987      1778 ...
5  |  3   54    333    1260     3600     8568     17934 ...
6  |  6  183   1620    8010    28275    80136    194628 ...
7  | 11  636   8208   53240   232500   785106   2213036 ...
8  | 23 2316  43188  366680  1979385  7960638  26037431 ...
9  | 46 8610 232947 2590420 17287050 82804806 314260765 ...
...

Andrew Howroyd, Table of n, a(n) for n = 1..1275
Virginia Perkins Johnson, Enumeration Results on Leaf Labeled Trees, Ph. D. Dissertation, Univ. South Carolina, 2012.

Columns 1..5 are A001190, A083563, A220816, A220817, A220818.
Main diagonal is A319580.
Cf. A241555, A319254, A319541.

A:= proc(n, k) option remember; `if`(n<2, k*n, `if`(n::odd, 0,
      (t-> t*(1-t)/2)(A(n/2, k)))+add(A(i, k)*A(n-i, k), i=1..n/2))
    end:
seq(seq(A(n, 1+d-n), n=1..d), d=1..12);  # Alois P. Heinz, Sep 23 2018

A[n_, k_] := A[n, k] = If[n<2, k*n, If[OddQ[n], 0, (#*(1-#)/2&)[A[n/2, k]]] + Sum[A[i, k]*A[n-i, k], {i, 1, n/2}]];
Table[A[n, d-n+1], {d, 1, 12}, {n, 1, d}] // Flatten (* Jean-François Alcover, Sep 02 2019, after Alois P. Heinz *)

\\ here R(n,k) gives k-th column as a vector.
R(n,k)={my(v=vector(n)); v[1]=k; for(n=2, n, v[n]=sum(j=1, (n-1)\2, v[j]*v[n-j]) + if(n%2, 0, binomial(v[n/2]+1, 2))); v}
{my(T=Mat(vector(8, k, R(8, k)~))); for(n=1, #T~, print(T[n,]))}

T(1,k) = k.
T(n,k) = (1/2)*([n mod 2 == 0]*T(n/2,k) + Sum_{j=1..n-1} T(j,k)*T(n-j,k)).
G.f. of k-th column R(x) satisfies R(k) = k*x + (R(x)^2 + R(x^2))/2.

A319376 Triangle read by rows: T(n,k) is the number of lone-child-avoiding rooted trees with n leaves of exactly k colors.

Original entry on oeis.org

1, 1, 1, 2, 6, 4, 5, 30, 51, 26, 12, 146, 474, 576, 236, 33, 719, 3950, 8572, 8060, 2752, 90, 3590, 31464, 108416, 175380, 134136, 39208, 261, 18283, 245916, 1262732, 3124650, 4014348, 2584568, 660032, 766, 94648, 1908858, 14047288, 49885320, 95715728, 101799712, 56555904, 12818912

Offset: 1

Andrew Howroyd, Sep 17 2018

Lone-child-avoiding rooted trees are also called planted series-reduced trees in some other sequences.

			Triangle begins:
    1;
    1,     1;
    2,     6,      4;
    5,    30,     51,      26;
   12,   146,    474,     576,     236;
   33,   719,   3950,    8572,    8060,   2752;
   90,  3590,  31464,  108416,  175380,  134136,   39208;
  261, 18283, 245916, 1262732, 3124650, 4014348, 2584568, 660032;
  ...
From _Gus Wiseman_, Dec 31 2020: (Start)
The 12 trees counted by row n = 3:
  (111)    (112)    (123)
  (1(11))  (122)    (1(23))
           (1(12))  (2(13))
           (1(22))  (3(12))
           (2(11))
           (2(12))
(End)

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (first 50 rows)

Columns k=1..2 are A000669, A319377.
Main diagonal is A000311.
Row sums are A316651.
Cf. A141610, A242249, A255517, A256064, A256068, A319254, A319541, A326396.
The unlabeled version, counting inequivalent leaf-colorings of lone-child-avoiding rooted trees, is A330465.
Lone-child-avoiding rooted trees are counted by A001678 (shifted left once).
Labeled lone-child-avoiding rooted trees are counted by A060356.
Matula-Goebel numbers of lone-child-avoiding rooted trees are A291636.
Cf. A000014, A000081, A000169, A005804, A206429, A330951.

b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(binomial(A(i, k)+j-1, j)*b(n-i*j, i-1, k), j=0..n/i)))
    end:
A:= (n, k)-> `if`(n<2, n*k, b(n, n-1, k)):
T:= (n, k)-> add(A(n, k-j)*(-1)^j*binomial(k, j), j=0..k-1):
seq(seq(T(n, k), k=1..n), n=1..10);  # Alois P. Heinz, Sep 18 2018

b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, Sum[Binomial[A[i, k] + j - 1, j] b[n - i j, i - 1, k], {j, 0, n/i}]]];
A[n_, k_] := If[n < 2, n k, b[n, n - 1, k]];
T[n_, k_] := Sum[(-1)^(k - i)*Binomial[k, i]*A[n, i], {i, 1, k}];
Table[T[n, k], {n, 1, 10}, {k, 1, n}] // Flatten (* Jean-François Alcover, Sep 24 2019, after Alois P. Heinz *)
sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
mtot[m_]:=Prepend[Join@@Table[Tuples[mtot/@p],{p,Select[mps[m],1Gus Wiseman, Dec 31 2020 *)

\\ here R(n,k) is k-th column of A319254.
EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
R(n, k)={my(v=[k]); for(n=2, n, v=concat(v, EulerT(concat(v, [0]))[n])); v}
M(n)={my(v=vector(n, k, R(n,k)~)); Mat(vector(n, k, sum(i=1, k, (-1)^(k-i)*binomial(k,i)*v[i])))}
{my(T=M(10)); for(n=1, #T~, print(T[n, ][1..n]))}

T(n,k) = Sum_{i=1..k} (-1)^(k-i)*binomial(k,i)*A319254(n,i).

Sum_{k=1..n} k * T(n,k) = A326396(n). - Alois P. Heinz, Sep 11 2019

A339779 Array read by antidiagonals: T(n,k) is the number of homeomorphically irreducible leaf colored trees with n leaves of k colors.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 3, 1, 0, 1, 4, 6, 4, 2, 0, 1, 5, 10, 10, 11, 3, 0, 1, 6, 15, 20, 36, 30, 7, 0, 1, 7, 21, 35, 90, 144, 105, 13, 0, 1, 8, 28, 56, 190, 476, 706, 380, 32, 0, 1, 9, 36, 84, 357, 1251, 3034, 3774, 1555, 73, 0, 1, 10, 45, 120, 616, 2814, 9845, 21380, 22140, 6650, 190, 0

Offset: 0

Andrew Howroyd, Dec 16 2020

Homeomorphically irreducible trees are trees without vertices of degree 2. All non-leaf nodes then have degree >= 3.
Not all colors need to be used.
The Johnson reference has a mistake in formula 4.3. In particular, the final term should be subtracted rather than added. Compare with the first formula given here. The table of results given in the reference is consequently also incorrect.

			Array begins:
============================================================
n\k| 0  1    2      3       4       5        6         7
---+--------------------------------------------------------
0  | 1  1    1      1       1       1        1         1 ...
1  | 0  1    2      3       4       5        6         7 ...
2  | 0  1    3      6      10      15       21        28 ...
3  | 0  1    4     10      20      35       56        84 ...
4  | 0  2   11     36      90     190      357       616 ...
5  | 0  3   30    144     476    1251     2814      5656 ...
6  | 0  7  105    706    3034    9845    26383     61572 ...
7  | 0 13  380   3774   21380   85995   274800    744556 ...
8  | 0 32 1555  22140  163670  812160  3086481   9692480 ...
9  | 0 73 6650 137096 1322960 8092945 36550458 132954360 ...
     ...

Andrew Howroyd, Table of n, a(n) for n = 0..1325
Virginia Perkins Johnson, Enumeration Results on Leaf Labeled Trees, Ph. D. Dissertation, Univ. South Carolina, 2012. [Table 4.3 is an incorrect version of this table].

Columns k=1..4 are A007827, A339782, A339783, A339784.

Cf. A319254 (planted), A339649 (degree <= 3), A339780.

\\ here R(n,k) is k-th column of A319254.
EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
R(n, k)={my(v=[k]); for(n=2, n, v=concat(v, EulerT(concat(v, [0]))[n])); v}
U(n, k)={my(g=x*Ser(R(n,k))); Vec(1 + g + k*x*g - g^2)}
{my(T=Mat(vector(9, k, U(8, k-1)~))); for(n=1, #T~, print(T[n, ]))}

T(n,k) = k*g(n-1,k) + g(n,k) - Sum_{j=1..n-1} g(j,k)*g(n-j,k) for n > 1 where g(n,k) is A319254(n,k).

G.f. of k-th column: 1 + k*x*r(x) + r(x) - r(x)^2 where r(x) is the g.f. of the k-th column of A319254.

A319369 Number of series-reduced rooted trees with n leaves of n colors.

Original entry on oeis.org

1, 3, 28, 430, 9376, 269675, 9632960, 411395268, 20445999734, 1159248404721, 73846864163348, 5221802726902476, 405858598184643930, 34392275731729465799, 3155760058245300968416, 311720334688779807141832, 32980137195294216968253900, 3720954854814866649904474180

Offset: 1

Andrew Howroyd, Sep 17 2018

nonn

Not all of the n colors need to be used.

Alois P. Heinz, Table of n, a(n) for n = 1..340
V. P. Johnson, Enumeration Results on Leaf Labeled Trees, Ph. D. Dissertation, Univ. Southern Calif., 2012.

Main diagonal of A319254.

Cf. A000311 (1 leaf of each color), A316651.

b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(binomial(A(i, k)+j-1, j)*b(n-i*j, i-1, k), j=0..n/i)))
    end:
A:= (n, k)-> `if`(n<2, n*k, b(n, n-1, k)):
a:= n-> A(n$2):
seq(a(n), n=1..20);  # Alois P. Heinz, Sep 18 2018

b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, Sum[Binomial[A[i, k] + j - 1, j]*b[n - i*j, i - 1, k], {j, 0, n/i}]]];
A[n_, k_] := If[n < 2, n*k, b[n, n - 1, k]];
a[n_] := A[n, n];
a /@ Range[1, 20] (* Jean-François Alcover, Sep 24 2019, after Alois P. Heinz *)

EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
a(n)={my(v=[n]); for(n=2, n, v=concat(v, EulerT(concat(v, [0]))[n])); v[n]}

a(n) ~ c * d^n * n^(n - 3/2), where d = 1/(2*log(2) - 1) = 2.588699449562089830805384431942090... and c = 0.2580000331300831455241033648... - Vaclav Kotesovec, Sep 18 2019, updated Mar 16 2024

cp's OEIS Frontend

A000669 Number of series-reduced planted trees with n leaves. Also the number of essentially series series-parallel networks with n edges; also the number of essentially parallel series-parallel networks with n edges.

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A050381 Number of series-reduced planted trees with n leaves of 2 colors.

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A319539 Array read by antidiagonals: T(n,k) is the number of binary rooted trees with n leaves of k colors and all non-leaf nodes having out-degree 2.

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A319376 Triangle read by rows: T(n,k) is the number of lone-child-avoiding rooted trees with n leaves of exactly k colors.

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A339779 Array read by antidiagonals: T(n,k) is the number of homeomorphically irreducible leaf colored trees with n leaves of k colors.

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A319369 Number of series-reduced rooted trees with n leaves of n colors.

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A220823 Number of rooted gene trees with n leaves on the label set [3].

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